AN APPROXIMATE SOLUTION FOR. 3 AIR-COUPLED RAYLEIGH' wAVEs . PROPOGATIN'G ACROSS AVERIICAL BOUNDARY . - l“l‘fl o 0 an! ' _ MICHlGAN STATE uuw ' . ER RAY TURPEMNG ' > 1972 '~ - . . . . . - . ,. . ‘ .l o c v o . . . - , . .. — ¢ “ . w , .. t . 1.! I. l ' . l . p ’ l .' . I v ': a ’ n u _‘ r . . - - . ‘ ' ' I. O , . ‘ . . . l _ . - I ‘ u . _. ’ I - . ‘ . , .r_ - u . . I O - , . I . n . A - . . - I . I . .‘ . . . . , . ’ . . h c . . . , . a . - - A I .-— I ‘ . . A I . n . .' r ' . . , ., ' f 1“ l - . ‘ n ' ' q 3 u - . , ' I. ‘ . _ ’ " .' i ' ‘ - ' I , . I _ . o - ~ 4 ‘ ' . ,I p . . . - I O o ' .. - ‘ , J - a 1 ' ' Iv . y I o - - . - . _ , ' . - ' . , ' ‘ l _ ‘ ’ .' ‘. , ' ' t . . . , . . . . , , I . . . ' a ' ‘ r , ' ‘ ' ‘_ I a . . . ." I ' a - o . l ’ ‘ ‘ . . ‘ ‘.I~- - -'..‘..".?...?V l- ————‘ . . — . .. -o I ‘ . ‘. . o o . I . . ; I - . . ‘A. . . . ._ v o ... ,. .. ~. ._ 'l- Ol- J o . .a . -._“-o . . o . - U 0.- .... . -oo oo ;. , .. 'l l . .- ~.v -. . ,n , It ‘ c - \ ~ - o . . .- . 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' ,'_ ‘mm 2' ABSTRACT AN APPROXIMATE SOLUTION FOR AIR-COUPLED RAYLEIGH WAVES PROPOGATING ACROSS A VERTICAL BOUNDARY By Walter Ray Turpening An approximate solution to the problem of the propaga- tion of air-coupled Rayleigh waves across a vertical bound- ary between two materials is obtained by solving the matrix of coefficients for the boundary and source conditions. Three cases are studied: 1) the boundary between two solids, 2) the boundary between a fluid and a solid with the source over the fluid and 3) the boundary between a solid and a fluid with the source over the solid. Gaussian Elimination is used to solve the matrices and determine the coefficients of the associated displacement potentials. The potentials in turn are used to find the expressions for the surface displacements as functions of the distance from ground zero and from the boundary. It is observed that compared with the displacements for a single material the boundary introduces attenuation and as increase in oscilla- tion of the associated Bessel functions for the region beyond the boundary. AN APPROXIMATE SOLUTION FOR AIR-COUPLED RAYLEIGH WAVES PROPOGATING ACROSS A VERTICAL BOUNDARY By Walter Ray Turpening A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTERS OF SCIENCE Department of Geology 1972 ACKNOWLEDGEMENTS I wish to thank Dr. Hugh F. Bennett for the continuing encouragement in the pursuit of the solution to this prob- lem, Dr. William J. Hinze for encouraging me to continue for a Masters of Science degree at Michigan State University. Dr. Roger M. Turpening. my brother, is most gratefully acknowledged for the suggestion of the problem and his guidance in my entire education. My parents are acknowledged for their undying faith in my ability to succeed in my studies. Many thanks to Karen Douglas for patient typing and helpful suggestions in the preparation of the final manu- script and Jesse Douglas for putting up with all of the preparations of the final manuscript. ii TABLE OF CONTENTS LIST OF FIGURES.................................... iv Notation........................................... v Introduction....................................... 1 The Problem........................................ The Solution... ..... ............................... The TheorleOOOOOOOOCOQOOOOOOOOOO-OOOOOOOOCO0.0 oommw Matrix Formulation-~General Case.............. Matrix Formulation--Case 2.................... 14 Matrix Formulation--Case 3.................... 21 Surface Displacements-........................ 27 Discussion and Conclusions......................... 34 Bibliography....................................... 36 ApmndiXOOOOOOOOOOOOIOOOOOOIOOOO...OOOOCOOOOOOOOOOO 38 iii Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES iv NOTATION p density a; P-wave velocity p; S-wave velocity fir dilatation potential %' rotational potential r. e . z cylindrical coordinates pij stress (i-th face. j-th direction) k radial wave number k axial wave number q radial displacement w vertical displacement a: angular frequency '}v# constants of Lame Jo(kr)' Bessel's functions (O-th and 1-st orders) J1(kr) It will be assumed that the potentials being used will be functions of r. z and time (t). The notation being used follows that of Jardetzky and Press (1952) where the axial wave numbers are defined as: vi z;0,r0.r7r INTRODUCTION This thesis is concerned with the phenomenon of Rayleigh waves coupled to atmospheric compressional waves and the effect of the change in surface material on the coupled Rayleigh waves. We will limit oursleves here to the case of the boundary between the two materials being per- pendicular to the horizontal surface. ,Page Figure 1. illustrates the physical situation. Lamb (1932) showed that air-coupled Rayleigh waves are generated in a dispersive medium when an air wave passes over the surface of the medium if the phase velocity of the Rayleigh wave at some frequency matches the speed of sound in air. Therefore. the characteristics of air-coupled Rayleigh waves are constant frequency and a phase velocity of sound in air. Jardetzky and Press (1952) developed the theory of air- coupled Rayleigh waves for the case of a solid surface layer overlying a solid half space. The thoery adds a branch to the dispersion curves for Rayleigh waves which illustrates the frequency dependence of air-coupling. Harkrider and Flinn (1970) considered the problem of a nuclear blast at high altitudes and the associated Rayleigh waves generated. Crustal structure and its effect on the Rayleigh waves at teleseismic distances were of primary interest. A realistic isothermally layered atmosphere. three different continental models and one oceanic model were considered. Their results showed that the oceanic structure gives rise to an order of magnitude higher ver- tical component versus the same conditions over continental models. Taking anelastic attenuation into account reverses the results at long distances. Much work has been done in the area of interpreting the information from the experimental records of sonic booms. Goforth and McDonald (1969, 1970), Espinosa, et al., (1967). Oliver and Isacks (1962) all considered the experimental problem of air-coupling. Other works too numerous to mention here are listed in the bibliography. THE PROBLEM We will be considering three different cases of the same configuration shown in Figure 1. The general case will be the situation of two solids in contact with their difference being the physical properties. The second case is of prime interest. Material I is a fluid and material II is a solid. Since the fluid has a shear modulus of zero (A: 0) the shear wave is nonexistant. therefore, the fluid does not support Rayleigh wave type propogation and no air-coupled Rayleigh wave occurs. This case is used to determine the generation of the air-coupled waves as a function of distance from the boundary. Interest in this case arises from the problem of recording an air shot near a river. lake or other body of water. Case three reverses the fluid in case two and will be used to study the reflection of the air generated waves. In all cases the main interest will be the radial and axial displacements of the solid surface as a function from ground zero. ... .___* ________________ l i I 11 Figure 1. A1 A2 _:____*__./ _____ / _____ d I \B \B I C C 1 __ 2 =0 I‘L / r-ro / Figure 2. THE SOLUTION The Theory Following the method of Lamb (190h) and Jardetzky and Press (1952) the horizontal and vertical components of dis- placement can be written: ’1 2 -21 3 q-arIJzar (1) a2 2 W=§g+fi+%%' (2) The stress components written: = m4 + 9%} p22 .. a a pzr ”#(an'+§_¥) a (3) prrfivzv‘Wfi with 44 J (A 2 _32 1o) :32 ‘7 ¢5§§H5¥5§ where r and z are cylindrical coordinates. In general each fgi and SL1 must be a solution to the reduced form of the wave equations: (v2 + (£25)¢i = 0 (4a) aH 1 and (72+ 9%)951 = 0 (4b) #1 Equations of the form given above were studied by Bessel and the solution is the well known form of the Bessel function: i = Aexp(iwtf7’z mom) (5) The factor exp(iat) will be assumed when not written to save space. To completely describe the functions. ¢i and ¢i above for some particular problem the boundary conditions and any initial conditions must be imposed. Following are the stress and displacement conditions that describe the prob- lem: at z = O, r1< ro (pzz)1 = (pzz)o (pzr)1 = 0 (6) w1 - wo at z = 0, r > r (7) IO N '1 V N ll 0 at 2‘7 0. r z r (pzz)2 (pzz)1 (1)21.)2 = (pzr)1 w2 ’ "1 q2 = q1 (8) Following Jardetzky and Press (1952) the plane through the source z = -d introduces the conditions of continuity of pressure across the plane and the discontinuity of the ver- tical component of velocity. given by: and respectively. )9th Iooat 3945 396 __ -—E - ——§ .. 2YJo(kr) (9) Using displacement potentials in the form of equation (5) the response materials and air can be described with respect to the source. for each of the specified regions: 7‘6 0 7‘1 (1 5‘2 AJo(kr)exp(Y;z) (Bexp( - 7’02 )+Cexp( 1’02) )Jo(kr) (DJo(kr)+D'Jo( k( ro-r) ) )exp( 412) (MJo(kr)+M'Jo(k(ro-r)))exp(-1éz) FJo(k(r-ro))exp(-féz) Below are the potentials to be used 2 < -d (10) -d< z< 0 (11) (12) (13) (1h) 3 72 = PJo(k(r-ro) )exp< - n+2) (15) with F' and F"-—-+ F and P' and P"-—- P At this point we must note that equations (12-15) are ap- proximations to the solution displacement potentials. Assuming that the boundary is reasonably flat compared to the curvature of the cylindrical wave front the reflected waves D' and M' can be cylindrical waves propogating from the boundary as indicated by the Jo(k(ro-r)) terms. Also in the exact case we must have the radial wave numbers equal across a horizontal surface and axial wave numbers equal across a vertical boundary. Since we will be con-‘ sidering the surface displacements the second approximation for the wave numbers can be neglected to symplify the anal- ysis. Figure 2 gives the boundaries and regions associated with each amplitude denoted by the primed and unprimed capital letters. Matrix Formulation-~General Case By substituting the potentials given in equations (10- 15) into the boundary conditions given by equations (6-9) the following equations can be obtained to solve for the amplitude functions. 2 2 Jo(kr)(-)1‘f§ + 2/111f)+D'JO(k(rO-r))(-k1:’§ + 2/17“?) 1 1 +MJO(kr)(-’)’2k2)+M'Jo(k(ro-r))(-721(2) (16) 2 +(B+Cl)Jo(kr)(7.O“—’§) = o “o DJO(kr)(—7/1)+D'Jo(k(rO-r))(-7/1)"’1V1Jo(kr)(k2) (17) +M-Jo+(B-cl)Jo(kr)<4;) = o 2 13.11(kr)(2k/1)+D'J1(k(ro-r))(-2k1’1)+M(-J1(kr))(2k1§+‘-"-§) Isl (18) 2 +M'J1(k(rO-r))(2k1§+ 9-5) = o 1 < )) >.£E-+ 12 +P ( >> 1 2 FJO(k r-ro (- Zaz 2M2 3) Jo(k r-ro (- 4k ) 2 (19) 02 +(B+CZ)JO(kr)()O;-2-) = o O FJO(k(r-ro))<-15)+PJO (20) +Jo<¢g> = o FJ1(k(r-ro))(2155/2) (21) 2 +PJ1(k(r-ro))(21fi+/‘:—’-§)(-k/«2) = 0 2 10 Dexp(- 7'12 )(2/117’1k)J1(kro) 2 ”2 +Mexp(-‘/zz)(27'z+ 7)(-kJ1(kro)) = 0 Fl (22) Dexp<-1’12)(7’1le(kro))+Mexp(-izz)(12kJ1(kro)) = o (23) Dexp( -7’12)(-7’1Jo(kro) )+D'exp(-1’12 )( ~11) +Mexp( - 1’22 )sz0( no )+M'exp( - {22 ) ( 1:2) +Fexp(-7’32)(~7’3)+Pexp(-7’uz)(-k2) = o 2 Dexp( - 112)[Jo(k15)( - 11:42- - 2/1k2)+2/1(r£-)J1(kro)1 O 1 2 -i-D'exp(-7’lz)[-31&2 4’ 2/41 “1 +Mexp( 422 )[Jo(kro)(2/u1 '7’2k2)-gu1 +M'exp( - 7/22) {-2/11 7/2( -k2+ 12"” 02 +Fexp( - 1’32 ) [12—2 - 2/12( -k d 2 +Pexp(-/u2) [2,12 /L}(-k2+ %)I = 0 (A1 ~01 )exp( - iod )-Bexp( 7’Od) = elm ° (A1 -C1 )exp( - /od)+Bexp( /od) 0 (12-02 )exp( - gm-Bexm 4,11) (AZ-C2)exp(-73d)+Bexp(10d) = __ ( -k2+ 5 J(kr)] 01 o (211) (25) (26) (27) (28) (29) 11 These equations can be reduced in number by eliminating four of the coefficients through the last four equations. These are associated with the conditions at the source plane. The result is: Yexp( - fad) B 1’0 Yexp( 13d) 01 = Al- 1% (30) Yexp(7gd) C2 = A2- 16 Using the above results we can write an augmented matrix of the coefficients. The lower case letters denote the coef- ficients that are funtions of the wave numbers and the con- stants of the various media. At this point we can note that the amplitudes of the potentials will be determined in terms of the amplitude of the source as given by the discontinuity of vertical velocity. Jardetzky and Press (1952) note that this difficulty can be generalized using the Fourier-Bessel integral by taking Y = k dk and integrating with respect to k from 0 to infinity. Proceeding in this direction by first solving the augmented matrix by Gaussian Elimination (this method can be found in any text on numerical methods. for example: Conte, p. 155ff). This method triangularizes the matrix. then by back substituting. each of the ampli- tudes is determined. Using the following relations obtained from the equa- tions for the M with Y2 boundary conditions: Dexp(122-71z) 7’2 if E N N #39 + MN (31) N 2; o 2 o 10 ) Y 7 YJO(kr)ZSinh(1od) Y6 -YJO(kr)ZCosh(76d) we can rearrange the augmented matrix of coefficients into the following: a7 a6 d7 d; m% Y7 d6 dé mé Y6 d5 d5 mg (32) du dd mi fa d3 d5 mi f3 f2 a2 Y2 f1 a1 Y1 Without boring the reader with all the steps in the method we will give the resulting triangularized matrix. interested the appendix. For those details of the reduction are given in the 13 a7 d7 d5 ”5 Y7 A A 0 A o A A A 5 d5 m5 Y5 Rd f“ In (33) f3 Y3 A 32 Y2 B 8L1 Y1 The various symbols used on the elements of the matrix are explained in the appendix. Now if we make all of the back substitutions the amplitudes of the displacement potentials are determined in terms of the Lame constants of the partic- ular mediums, wave numbers, and the source. The reduced potentials become: R5 YJo(kr)exp(iszQ1 (3“) YJo(kr)exp('loz)JK2 exp( - 'I’o(z+d)) #3 = YJo(kr)I exp(i d) + exp('/oz)(A1- ——'—°—-)] 0 40 (35) exp(-1’ (z+d)) exm 'l d) :0 + exp< 102x212- ——;;9-—>] 5’1 = YJO(kr)exp(-1’1z)(-2exp(-1’od))[A 4' J0(11:(ro-r))l\z:_)3 YJo(kr)[ 6) 14 (1 YJO( kr)exp( - 7’22)(-2exp( - iod)) (37) exp(z( - )) [A 2 1 *M'Jowuo'flfl YJO( kr)Jo( k( r-ro) )exp( - 132 ) ( ~2exp( -1’od) )I' (38) 22 5’2 27’ YJO( kr)Jo( k( r-ro) )exp( - qu ) ( -zexp( - 40d))—-1-I‘ NEW—3 /’2 (39) The capital letters used here are functions of the con- stants that arise from the reduction and solution of the augmented matrix. Each is shown in detail in the appendix. Matrix Formulation--Case 2 As was stated in the outline of the problem this case is one of the generation of an air-coupled Rayleigh wave in media II of Figure 1 with media I being a fluid. We can proceed with the same method as that was used in the general case and forming an augmented matrix from the equations for the boundary conditions as was done in the general case. Without hesitation we can write the boundary conditions using the same notation: z - O, r ¢ r0 (#0) 15 o (pzz)2 = (pzz)o (pzr)2 = o (#1) w2 ' W0 2 > O, r = r0 (prr)2 = (prr)1 (pzr)2 = 0 (#2) Q2 ‘ q2 z = -d 2' _ 91‘” P6&5% ' ft“% (#3) 345 MS ‘5'; - T'z' = ZYJo(kr) Again we can use displacement potentials to describe the response of the media in question to the source. Re- maining in cylindrical coordinates we have the potentials of the form of equation 5: ,5") = AJ°(kr)exp( 7’02) ((+11) 3 = (Bexp(-752) + Cexp(1gz))Jo(kr) (45) 21 = (DJo(kr) + D'Jo(k(ro-r)))exp(-7’lz) (46) 52’ 2 = FJ°(k(r-ro))exp(-7’32) (47) fife = PJo(k(r-ro))exp(-¥Lz) (#8) 16 The reader is referred to Figure 3 for the relative posi- tion of the amplitudes of equations (44-h8) denoted by the capital letters. Similar to the method of the general case we insert the potentials of equations (nu-48) into the boundary conditions and rearranging to the following equations. w? a? DJo(kr)(-)I:§) + D'Jo(k(rO-r))(-)1;§) 1 1 2 (49) + (B+cl)Jo(kr)()o£'-’-§) = o o DJo(kr)(-41) + D'Jo(k(ro-r))(-4&) (50) + (B-cl)Jo(kr)(¢g) = o 2 Dexp(-4lz)Jo(kro)(-X1f§) + D'exp(-4lz)(-11:’-2- 1 1 2 + Fexp(-13z)()2§% - gy2(-k2+ %J)+Pexp(-/Lz) (51) 2 (2/24u(-k2+%)) = o DJ1(kro)exp(-7&z) + D'exp(-f12) * Fexp(-15z) (52) * Pexp('7u2)("u)= O Fexp(-'/32)(2,a2k7’3) + Pexp(-1’uz)(-/~2k(2'/fi+-w§)) = 0 (53) 2 17 Figure 3 Figure A 18 FJ1(k(r-ro))(2/a2k 13) (54) + 2 ya PJ1(k(r-ro))(jazk(21u+ -§)) = o 2 FJo(k(r-ro))(-)2;—’-2- +2/a21§) + PJo(k(r-ro))(-‘/uk2) ' 2 (55) 2 + (B+c2)Jo(kr)(xo£’§) = 0 '.° FJo(k(r-ro))(-7'3) + PJO(k(r-ro))(k2) (56) + (B-cz)Jo(kr)(«/o) = o with the same source equations as in the general case. equations (26-29). Reducing the number of equations as we did in the general case we have: ”2 wz (0—2 DJO(kr)(-11;§)+D'Jo(k(ro-r))(-)~1;-2-)*A1Jo(kr)()o ) 1 1 “o ( ) 57 1.23 ouz = YJo(kr)2sinh(dgd) ¢o° DJo(kr)(-"/1)+D'Jo(k(ro-r))(-’7’1)+A1J0(kr)(-"/o) (58) = -2YJo(kr)cosh( 10d) 19 2 2 Dexp( - 7’12)Jo(kro)(-7(19-§)+D'exp(- ilz)(-le'§) “1 1 2 + F[6XP(-7’3z)('>129’-§ - 2.21-1.24 1;) )texp(-fuz) (59) K 2 11 7’7’ (-k2+1‘-) (/«2le 2)]-o 2 2124-2- 1+ '6: DJO(kro)exp(-/lz)+D'exp(-7’12) (60) 2 + F[exp(-¢32)-exp( 7’42)(-/-3—ZL-’——)] = 0 Ki 2 L1 2 /’2 2 27’ /k2 2 FJo(k(r-ro))(-)2:—’§ + 2,57% - fimzaommo?) 2 2¥u+~§ O (‘2 (61) 2 )0“: = YJo(kr)Zsinh(/od) {0° 2/1:2 FJo(k(r-ro))[-/3*—;;l—;2- 4‘ A2J0(kr)(-7’o) 2 +— u 2 (62) #2 = -YJo(kr)2cosh( 70d) with B Yexp(~7’od) A Yexp(/od) 7’0 1 1 7’0 20 Y ( d) 2/ C =A - exp 7’0 P=F(-—-l-—) 2 2 7’0 272+£ 1+ 2 lflz Forming the augmented matrix of coefficients and applying the Gaussian elimination method we have the following tri- angularized matrix: a1 d1 di Y1 A A . A d2 d2 Y2 . A A d3 f3 Y3 (63) A A f L1 Yu A as Y5 A 3‘6 Y6 Back substitution obtains the amplitudes which we use to get the following displacement potentials. 1A1 94") = YJo(kr)exp( 7’02) ‘02 X»? 0 (6+) = YJo(kr)exp( 7’02 )—-‘%1——- ()5f§-¢g) O 21 exp(- 70(ztd)) 7’0 %" = YJo(kr) epo d) + exp(7’oz)( A12 - +) w \0—2 “0 (65) eXp(-1'o(z+d)) ’I’o YJO(kr) A 1 + exp< 7’oz)(-—§-2-— - ————9—) ) @Ln./ 0‘2 0 ‘S\ H II YJo(kr)exp(- 7’lz)( ~2exp( - 'r’od) {A +A'Jo( k(ro-r) )1 (66) his A) u YJO( kr)Jo( k( r-ro) )exp( 4’32 )(-26xp( - 70d) )F (67) \QL n) (I YJo(kr)Jo(k(r-ro) )exp(-iuzx-zexpe ion)? (68) Matrix Formulation -- Case 3 Without being repetitive we can proceed as before and write directly the results of similar calculations that were done in the previous cases. Keeping in mind that a fluid does not support shear wave propogation and therefore no Rayleigh waves. The boundary conditions for this case are as follows: N u f” H A n The displacement potentials to be used here are as follows: 22 *d N "S v v H II 1 £ II A "U N N v N I V V II II .0 H II AJo(kr)exp(1BZ) (Bexp(-W52)+Cexp(1gz))Jo(kr) (DJo(kr)+D'Jo(k(ro-r)))exp(-iiz) (MJo(kr)+M'Jo(k(rO-r) ) )exp(- /2z) #3 WOT; 2YJo(kr) (69) (70) (71) (72) (73) (74) (75) (76) 23 %2 = FJo(k(r-ro))exp(-iéz) (77) The reader is referred to Figure h for the relative position of the potentials. Inserting the potentials into the bound- ary conditions we have the following equations. DJ (kr)(— “—Z-rz 42)+D'J (k(r -r))(- 93+ 1’2) 0 110‘2 ”1 1 o 0 3‘10‘2 9’3 1 1 1 + MJo(kr)(-‘7’1k2) + M'Jo(k(ro-r))(-v’2k2) (78) w? + (B+C1)Jo(kr)(lo-§) = O o DJo(kr)(-¢1) + D'Jo(k(rO-r))(-¢3) + MJo(kr)(k2) (79) + M'J°(k(ro-r))(k2) + (B-Cl)Jo(kr)(-7’o) = o DJ1(1cr)(2k¢1)+D'J1(k(ro-r))(-211k) (80) + MJ (kr)(-2k/2+£P—-) + M'J (k(r -r))(2k12+9-2-) =0 1 2 2 o o 2 2 '61 ’51 FJo(k(r—ro))(-’Al;g) 4- (B+C2)Jo(kr)(1o:§) = o (81) O FJo(k(r-ro))(-7’3) + (B-02)Jo(kr)(‘/o) = o (82) 21+ “2 Fexp(- 73:2)(12—2- 2)* Dexp(-112)(Jo(kro)(-11- - 2k2,u,) “2 “1 +J1(kro)2,«1;k;) + D'exp(-'/lz) 2 (a 2 k 1 + Mexp( - 7’2z )(-2»17’2)(-k2Jo(kro)+ f—J1(kro)) o + M'exp(-1’Zz)(-2/A1)/2)(-k2- g) = o Dexp(-11z)(-2,u1‘/1) + Mexp(-1'22)(2/2+w—)= (81)) [51 Dexp(-7’12)+ Mexp(-'r’zzH-y/z) = 0 (85) Reducing the order of the equations we have the following 2 DJo(kr)(-').1“22- + gul'if-kz) + D'Jo(k(rO-r))(Jl‘x2 ‘32—:- —-+2,u1-/2) 1 “1 + M'J (k( ))( 9’k2)+A J (k )( i) (86) o ro-r ' 2 1 o r X0 2 O 2 6) )0“? = YJo(kr)Zsinh( 70d) {0° k2 2 DJ o(kr)(7; -/1) + D'Jo(k(ro-r)) + M'Jo(k(ro-r))(k ) (87) +A1Jo(kr)(-/o) = -YJo(kr)2cosh(I/od) 2‘3 2 L DJ1(kr)(2k'/1-2k1'2- 0;", ) 4- D'J1(k(ro-r))(-2k7’1) I’i 2 + M'J1(k(ro-r))(2k¢§+ £5) fll =0 2 2 Fexp(-7’3z)()2 “’ —)+Dexp(- 7’loz)J (kro )(- )12“) 2"‘21"‘1 2 + D ' exp( - 7’1z)( -711‘-’-§ 1-2/1 ( Jig-122)) a 1 + M'exp( - 7’22 ) ( -2/a1 7/2)(-k2-%) = FJ (k(r-r ))(-12 :)+ A2 J o(kr)() —) o o 2012 OK?) 2 10%- O = YJo(kr)Zsinh( 9’Od) O FJo(k(r-ro))(-v’3) + A2J0(kr)(-1’o) = -yJo(kr)2cosh(¢;d) with ( 10d) B = Yexp—-———— “/o d C = A - (Yexp('/Q )) 1 1 7/0 (88) (89) (90) (91) Writing solving 26 Yexp<¢gd) c2 = A2- ( 4° ) Dexp(-Viz) 3 ll 7’2exp( ~¢zz) into an augmented matrix of the coefficients and we have the displacement potentials: = YJo(kr)exp(¢gz) 1 (u-gz) 0 do (92) YJ°(kr)exp(¢sz) exp( - 10(z'td) ) ’J’o YJO(kr) + exp(4gz) (.A1 exp(4gd)) ’)’o + lam ‘\ °a ON (93) exp( -'/0( z+d)) 7’0 YJo(kr) A2 + exp( 10d) ) + exp(igz)( / O 27 711 = YJo(kr)exp(-1’lziA +A'Jo(k(ro-r))} (94) exp(?’ z-7’ z) )1 ... 1.0.1.).1..-72.(1__72.;_1__ + 1.01.11-26.16.) :6 2 = YJo] mam} w1 = YJo(kr)(-Zexp(-1od)) (106) 2 7’ 2] -7/14- iii-)1}- [41A'+(1+¥§)Z"[']Jo(k( ro-r) )§ 29 q2 = Y2kexp(-1od) (107) TZJ1(kr)Jo(k(r-ro)) + Jo(kr)J1(k(r-ro))} w2 = YJo(kr)Jo(k(r-ro))(-2exp(-9’od))F(-i3) (108) For equations 97, 101, 105 the reader is referred to Figure 7, equations 98, 102, 106 to Figure 6, equations 99, 103, 107 to Figure 8 and equations 100, 104, 108 to Figure 5. 3O .m magmas \/ 0 xi \ / \u m u E / \ // \ 2/ \ / x \ Helix/K \ .. / 2 \ \ / 72.x X i \ , / \\ /.\ / ( / \\/..\ q, o / / \ <\ /\ \ (\\ m u ops I“. m u one I!!! N n Chum Illlla'Il-I O 0 0 ll}. 2 out: 2.51:. 0.“ Au CBC H. pawsmomaamwa Hmoflpum> H 31 N ll 0 fix .0 mmswwm / i\ \/ Ilaunll. o A“- svxvoslrxvos .l.l.ll 2505 on v a pcmEmomHmmHQ Hoowppm> 32 ex he .s mnsmfim -I Afioo-ovxvflshuxvos + 25$ Aos-uvxvosgsxvas OHAH pCoEmomHmmHQ Hawomm DISCUSSION AND CONCLUSION In this thesis we are looking at the surface displace- ments as a function of distance from ground zero and from the vertical boundary. If we consider the problem of a single material the vertical displacement has a distance dependence of Jo(kr) and the radial displacement has the distance dependence of J1(kr). Introducing a second mate- rial with a vertical boundary produces an additional factor in the potentials that indicates a wave starting at the boundary and propogating outward. The new factor is Jo(k(r-ro)). The distance dependence factor associated with the vertical and radial displacements are POUCr)Jo(k(r-ro))‘\and‘J1(kr)Jo(k(r-ro))+ Jo(kr) J1(k(r-ro)f]respectively. Refering to Figures 5 and 7 the effects of a boundary are clearly illustrated. The first effect is the distinct attenuation of the wave as compared with Jo(kr) and J1(kr) functions. Secondly there is an increase in oscillations per unit distance beyond the boundary in relation to the single material case for the same absolute distance from ground zero. Obviously only two frequencies are illustrated in the figures but there seems to be no reason to not expect the 3L» 35 same relative effect to be exibited for all frequencies. It appears that the boundary may have the effect of dis- torting the wave to a more complex form. Within the boundary, Figures 6 and 8 illustrate similar features of attenuation of the waves and increased oscilla- tion per unit distance. However, as has been stated earlier in this thesis the conditions and potentials are not com- pletely exact and the results as shown may be even more complex when studied more closely. BIBLIOGRAPHY BIBLIOGRAPHY Arfken, G.. Mathematical Methods for Physicists, 2nd ed., 815pp.. Academic Press, New York, 1970. Bateman. H.. Rayleigh waves, Proc. Nat. Acad. Sci. U.S., 24, 315-320. 1938. 133th, M., Mathematical Aspects g; Seismology, 415pp.. Elsevier. New York, 19 . Conte, 8.. Elementary Numerical Analygis. 278pp.. McGraw— Hill, New York, 19 5. Espinosa, A.F.. Sierra, P.J.. and Mickey. W.V.. Seismic waves generated by sonic booms: A geoacoustical problem. J. Acoust. Soc. Amer.. 4“, 1074-1082. 1967. Ewing, M., Jardetzky, W.S.. and Press, F., Elastic Waves in Layered Media. 380pp., MoGraw-Hill, New York, 1957. Goforth, T.T., and McDonald, J.A., A physical interpreta- tion of seismic waves induced by sonic booms. J. Geophys. Res.. 75. 5087-5092. 1970. Harkrider, D.G., and Flinn, E.A.. Effect of crustal struc- ture on Rayleigh waves generated by atmospheric explo- sions, Review of Geophy; and Space Phy., 8, 501-516, 1970. Jeffreys. H.. The Earth, 5th ed.. 525pp., Cambridge Univer- sity Press, New York. 1970. Kraut, E.A.. Fundamentals of Mathematical Phygics. h6upp., McGraw-Hill, New York. 19 7. Lamb, H.. On the propogation of tremors over the surface of an Elastic solid, Phil. Trans. Roy. Soc. A. 203. 1-“2. 190 . ----- , Hydrodynamics. 738 pp.. Cambridge University Press, Cambridge, England. 1932. Lindsay, R.B.. Mechanical Radiation, 415 pp.. McGraw-Hill, New York, 1960. ' 3.4 J 37 Love, A. E. H.. A Treatise on the Mathematical Theory of E1asticity.5th ed.,_6h3 pp.. Cambridge University Press. Cambridge, England, 1927. Morse, P.M.. Vibration and Sound, 2nd ed.. 468 pp.. McGraw- Hill, New-York." '197578‘.‘ Officer. C. 3.. Introduction to the Theory of Sound Trans- mission, 285 pp.. McGraw-Hill, New York, 1958. Oliver. J., and Isacks,B.. Seismic waves coupled to sonic booms, Geophysics. 27. 528-530, 1962. Press, F.. and Ewing. M”,Theory of air-coupled flexural waves, J. App. Physics, 22, 892-899, 1951a. ----- , Ground coupling to atmospheric compressional waves, Geophysics. 16- #30, 1951b. Press. F.. and Oliver. J.. Model study of air-coupled surface waves. J. Acoust. Soc. Amer., 27, 43-46,1955. APPENDIX APPENDIX GENERAL CASE Given the following equations for the boundary condi- tions for the interfaces of the solids: 2 2 a: 2 . to 2 DJO(kr)(-31;-§ + 23141)+D Jo(k(ro-r))(-)1;‘2 + 23.141) 1 1 +MJo(kr)(- 12k2)+M‘Jo(k(ro-r))(- 121(2) 02 +(B+C1)Jo(kr)()o-§) = o “o DJo(kr)(-4&)+D'Jo(k(rO-r))(-73)+MJo(kr)(k2) +M'Jo(k(rO-r))(k2)+(B-Cl)Jo(kr)(7g) = o DJl(kr)(2kfl)+D'J1(k(ro-r))(-2k1&)+M(-J1(kr))(2k1§j§§) 1 +M'J1(k(ro-r))(2k1§+ 9;) = 0 1 ) )é 3+ /2 FJo(k(r-ro) (- sz + 2»? 3) PJO(k(r-ro))(- 4k ) 2 +(B+CZ)JO(kr)(3.&£) = o o 2 “2 38 39 FJo(k(r-ro>)<-qg)+PJoJo = FJ1(k(r-ro))(275§w2) 12+ ”2) )- o +PJ1(k(r-ro))(2 4 ~5- (-k/12 - - 2 Dexp(-4’1z)(2/141k)J1(kro) 12+ ”2 1 +Mexp(-4zz)(22 —)( -kJ1(kro)) = O Dexp(-112)(')’1kJ1(kro) )+Mexp(-/Zz)( isz1(kro)) = Dexp(-1lz)(-'/1Jo(kro) )+D'exp(-’/lz)(-7’1 ) +Mexp(-7%z)k2Jo(kro)+M'exp(-7gz)(k2) +Fexp(~432)(-75)+PeXP(-7LZ)(-k2) = (02 Dexp(-11m (kr o)(-71:2 - 2/1k2)+2/41(-)J1(kro)] 1 2 +D'exp(~/1z)1-7\1‘:’-2- 1' 2u1(-k2"' 129)] 1 +Mexp( 422 )[Jo( kro ) ( 2,441 721:2 )-2/ul7’2;15<-)-J1 ( 1:15)] +M'exp(-¥22)[-2/I/é(-k2+ €§01 #0 02 +Fexp(-7§z)[)2:2 - 2/u2(-k2*'12£)} 2 “2712..-“. 5-1 = 0 From the conditions for the source plane we have: Yexp( 10d) B = ’10 _ A Yexpcyod) C - -( ’10 where the relation between A and C is the same over either material and kept separate by the subscripts 1 or 2. The differences arise from the different materials that reflect the wave associated with C (i.e. the -z propogating term written exp(7gz)). Making the substitutions for B and C1 and C2 we have the following: N 2 DJO(kr)(-)1:3§+ 2/17'2)+D J oo(k(r -r))(-71:—2- + 2/1/2) 1 1 MJo(kr)(-7ék2)+M'Jo(k(ro-r))(-¥2k2) 0:2 +(B+Cl )Jo (kr)()O 7:) = 0 d0 L11 DJo(kr)(- f1)+D'Jo(k(ro-r))(-1’1)+MJO(kr)(k2) +M'Jo(k(ro-r))(k2)+(B-cl)Jo(krY7g)= o 23) 2 1 DJ1(kr)(2k’/1)+D'J1(k(ro-r))+M(-J1(kr))(Zkfg-r 2 +M'J1(k(rO-r))(2k1'§+ 912-) = o 1 2 FJo(k(r-ro))(-52f% + gAng) + PJo(k(r-ro))(-7;uk2 2 +(131-c2 )Jo (kr)(df-’—- -—-:=) (X0 FJO(k(r-r0))(-7§)+PJO(k(r-ro))(k2)*(B-02)Jo(kr)(75) = 12%): 2 F( -21’3 )+P(2 D(2p1¥i)exp(-1iz)(-kJ1(kro)) 2 +M<2f§+ £25-)exp(-1’22)(-1+D'Jo)(-i1) 6 +M'Jo(k(r0-r)) =-¥Jo(kr)2cosh(7gd) ) l"2+1: ))“Z 2+ /2) A1J0(kr ()0-2) Jo(kr (- %:2 -k %”l 1 o 1 Hr (M ))( ) 93+ 7’2) Jo r'ofl? ' 1&2 91 2 7 1 “,2 07 O 0 +M'Jo(k(ro-r))(-71k2) = YJO(kr) Zsinh(1gd) Using lower case letters to denote the coefficients of the amplitudes we can write the previous equations in a more compact form: A28.1 + Ff1 = Y1 A2212 + Ff2 = Y2 +II+II+ Dd3 D d3 M m Ff Dd“ + D'dd * M'ma + Ff“ = 0 ll 0 Dd + D'd' + M' ' = 5 5 m 0 1+7 A136 + Dd6 ‘r D'dé + M'Mé Y6 A1a7 + Dd7 + D'd% + M'm% = Y7 Obviously the above system of equations is not homogeneous. The right side of some of the equations contains a term that is associated with the source. The method of solving these simultanous equations for the amplitudes specifically is by solving the nonhomogeneous matrix of amplitude coefficients, in other words an augmented matrix of coefficients. The matrix is written: a7 d7 dé m5 Y7 a6 d6 dé mé Y6 d5 d5 mg d” dd m& f“ d3 d5 m5 f3 f2 a2 Y2 f1 a1 Y1 At this point we can turn to numerical analysis for a method to solve the augmented matrix. Much work in this field of mathematics has gone into various ways to solve high order matrices with all sorts of manipulations to side step pro- blems that arise from using high speed computers. There are two basic problems that bother these computers, 1) multiply- ing by infinity and 2) dividing by zero, or numbers very 48 large or small respectively. Therefore. one must take care in choosing his method when using our handy machines. In the problem here we are trying to develOpe an algerbraic expression for the solution, thus we can use an extension of the old three equation--three unknown method from algerbra, Gaussian Elimination. One other factor is also helpful in our choice, the order of our matrix is only 7. Elements of the matrix to the left and below the prin- ciple diagonal are eliminated by multiplying an above row by the appropriate constant and subtracting the two rows. An illustration is given below: (Using the first three rows of the matrix for the illustra- tion.) a7 d7 d% m% Y7 d5 d5 m5 a7 d7 d% m% Y7 A ’ A A d6 dé ”é Y6 d5 d5 m5 with A * d6 = d6 - d7 a6 A . _ ' . * d6 ” d6 ‘ d7 a6 3 > o\. H a O\. I 3 \1 . SJ 0\ and then with and Q) m- 3) U1. D-<> 5 * d5 With back Y6 - Y7 a6 36. 8‘7 d7 (1,; A A. d6 d6 3. 5 d5 - dé d mg - fig d A '4:- -Y6 d5 d a? u\* vx* B> 3> S \n- 0w ~w- substitution matrix for a few amplitudes complex nature) with A 1 A1 = Jo(kr)[ ’70 l+< I‘m H 1 O 01 ON ) O O‘NIBN 49 in the original triangularized we have: (to illustrate the 2 sinh(¥6d) + 2exp< - 10d )[Ad7+A'd:,+n'm,;]] 5o and _ _JL_ - Y Cl .. 2A1 7—exp(7’od) u ‘3— ° 0 2 “o similarily _ Y A2 " ”2 A2 107 “/0 0(o with 1 2.2. 00(2 — . 0 A2 — J0(kr)[251nh(')’od) 7,0 - ZCOSh(7/od) (f -f ) —Jo(kr)(-2exp(-¥od))(d* 1f82 )] MICHIGAN STATE UNIVERSITY LIBRARIES : m IIIHHUIIIII ' 3 1293 03178 796